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Properties

Label 2-819-117.2-c1-0-54
Degree $2$
Conductor $819$
Sign $0.962 - 0.270i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.888 + 0.888i)2-s + (1.57 − 0.721i)3-s − 0.419i·4-s + (−0.372 + 1.39i)5-s + (2.04 + 0.758i)6-s + (−0.258 + 0.965i)7-s + (2.15 − 2.15i)8-s + (1.95 − 2.27i)9-s + (−1.56 + 0.904i)10-s + (1.20 − 1.20i)11-s + (−0.302 − 0.660i)12-s + (1.40 + 3.31i)13-s + (−1.08 + 0.628i)14-s + (0.416 + 2.45i)15-s + 2.98·16-s + (−1.57 + 2.73i)17-s + ⋯
L(s)  = 1  + (0.628 + 0.628i)2-s + (0.909 − 0.416i)3-s − 0.209i·4-s + (−0.166 + 0.621i)5-s + (0.833 + 0.309i)6-s + (−0.0978 + 0.365i)7-s + (0.760 − 0.760i)8-s + (0.653 − 0.757i)9-s + (−0.495 + 0.286i)10-s + (0.362 − 0.362i)11-s + (−0.0873 − 0.190i)12-s + (0.390 + 0.920i)13-s + (−0.290 + 0.167i)14-s + (0.107 + 0.634i)15-s + 0.746·16-s + (−0.382 + 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.00183 + 0.414066i\)
\(L(\frac12)\) \(\approx\) \(3.00183 + 0.414066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.57 + 0.721i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-1.40 - 3.31i)T \)
good2 \( 1 + (-0.888 - 0.888i)T + 2iT^{2} \)
5 \( 1 + (0.372 - 1.39i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.20 + 1.20i)T - 11iT^{2} \)
17 \( 1 + (1.57 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.606 + 2.26i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.46 + 7.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.57iT - 29T^{2} \)
31 \( 1 + (0.0162 + 0.00434i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.99 + 7.43i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (9.92 - 2.65i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.74 - 1.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.90 + 7.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 + (6.53 - 6.53i)T - 59iT^{2} \)
61 \( 1 + (-4.80 - 8.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.43 + 9.09i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.4 - 3.06i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.44 + 3.44i)T + 73iT^{2} \)
79 \( 1 + (-2.56 + 4.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.14 + 2.18i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.8 + 2.90i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.44 - 1.45i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.36064826404850957557774144030, −8.977689486152618995059631385049, −8.783012995475599779498052650003, −7.36246877791677686908599875161, −6.71847596933836950477407504633, −6.25049972394715446356281214426, −4.85624990397337247776943073536, −3.90256659559597822527427458454, −2.86182398350608576293537594098, −1.48112175474504686953904609257, 1.56862502186362962343153044216, 2.92227674096049393994088270379, 3.65875498109852135963554718536, 4.54645235436918043176861529886, 5.26973254133968234210048048987, 6.92147594709102421700623496102, 7.937489531963047130420992372669, 8.383677113677547596959036756640, 9.462523529107722077602023993566, 10.15442886091305538800943920748

Graph of the $Z$-function along the critical line